Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

number of possible Slater determinants a subset of most significant ones. Such
technique, computationally very heavy, is used in quantum chemistry to get
high-precision results in small molecules. Other, less heavy methods (the socalled
Møller-Plesset, MP, approaches) rely on perturbation theory to yield a
rather good estimate of correlation energy. A completely different approach,
which produces equations that are reminiscent of Hartree-Fock equations, is
Density-Functional Theory (DFT), much used in condensed-matter physics.
7.1.3 The Helium atom
The solution of Hartree-Fock equations in atoms also commonly uses the central
field approximation. This allows to factorize Eqs.(7.11) into a radial and
an angular part, and to classify the solution with the ”traditional” quantum
numbers n, l, m.
For He, the Hartree-Fock equations, Eq.(7.11), reduce to
∫
− ¯h2 ∇ 2
2m
1φ 1 (1) − Zq2 e
φ 1 (1) +
e r 1
∫
+
φ ∗ 1(2) q2 e
r 12
[φ 1 (2)φ 1 (1) − φ 1 (2)φ 1 (1)] dv 2
φ ∗ 2(2) q2 e
r 12
[φ 2 (2)φ 1 (1) − δ(σ 1 , σ 2 )φ 1 (2)φ 2 (1)] dv 2 = ɛ 1 φ 1 (1) (7.19)
Since the integrand in the first integral is zero,
∫
− ¯h2 ∇ 2
2m
1φ 1 (1) − Zq2 e
φ 1 (1) + φ ∗
e r
2(2) q2 e
[φ 2 (2)φ 1 (1)
1 r 12
−δ(σ 1 , σ 2 )φ 1 (2)φ 2 (1)] dv 2 = ɛ 1 φ 1 (1). (7.20)
In the ground state, the two electrons have opposite spin (δ(σ 1 , σ 2 ) = 0) and
occupy the same spherically symmetric orbital (that is: φ 1 and φ 2 are the same
function, φ). This means that the Hartree-Fock equation, Eq.(7.20), for the
ground state is the same as the Hartree equation, Eq.(6.19). In fact, the two
electrons have opposite spin and there is thus no exchange.
In general, one speaks of Restricted Hartree-Fock (RHF) for the frequent
case in which all orbitals are present in pairs, formed by a same function of r,
multiplied by spin functions of opposite spin. In the following, this will always
be the case.
7.2 Code: helium hf gauss
The radial solution of Hartree-Fock equations is possible only for atoms or in
some model systems. In most cases the solution is found by expanding on a
suitable basis set, in analogy with the variational method.
Let us re-write the Hartree-Fock equation – for the restricted case, i.e. no
total spin – in the following form:
Fφ k = ɛ k φ k , k = 1, . . . , N/2 (7.21)
The index k labels the coordinate parts of the orbitals; for each k there is
a spin-up and a spin-down orbital. F is called the Fock operator. It is of
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